On the control invariants of planar Bezier curves for the groups M(2) and SM(2)

Abstract

Let G=M(2) be the group generated by all orthogonal transformations and translations of the 2-dimensional Euclidean space E2 or G=SM(2) be the subgroup of M(2) generated by rotations and translations of E2. In this paper, global G-invariants of plane Bezier curves in E2 are introduced. Using complex numbers and the global G-invariants of a plane B curves, for given two plane B curves x(t) and y(t), evident forms of all transformations g\in G, carrying x(t) to y(t), are obtained. Similar results are given for plane polynomial curves.